Rational and irrational numbers | Year 9

01 / LEARN

Understand the idea.

What makes a number rational?

A rational number has an exact form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ . This includes integers: $7 = \frac{7}{1}$ . A terminating decimal is rational because its place value gives a denominator such as $10$ , $100$ or $1000$ .

A recurring decimal is also rational, even though it never ends. Its repeating block allows an exact fraction. An irrational number has neither a terminating nor a recurring decimal expansion; a short calculator display alone cannot establish either pattern.

a / b, a, b \in ℤ, b \neq 0

Square roots and pi

For a whole number $n \geq 0$ , $\sqrt{n}$ is rational exactly when $n$ is a perfect square. First ask whether an integer multiplied by itself gives $n$ ; the presence of a root sign does not settle the classification.

The exact number $π$ is irrational. A rounded value such as $3.14 = \frac{314}{100}$ is rational. An approximation can be useful without sharing the classification of the exact number it represents.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A root sign does not decide the answer

Classify $\sqrt{49}$ and $\sqrt{50}$ .

Worked solution

1
Simplify the perfect-square root.
$\sqrt{49} = 7 = \frac{7}{1}$
An integer is a ratio of two integers, so this value is rational.
2
Check whether 50 is a perfect square.
$7^{2} = 49 < 50 < 64 = 8^{2}$
There is no integer between 7 and 8 whose square is 50.
3
Classify the second root.
$\sqrt{50} is irrational$
A square root of a positive non-square integer is irrational.

Answer

$\sqrt{49}$ is rational; $\sqrt{50}$ is irrational.

EXAMPLE 02

An exact recurring decimal

Write $0. \bar{18}$ as a fraction and classify it.

Worked solution

1
Name the recurring decimal.
$x = 0.181818 \dots$
The two-digit block 18 repeats indefinitely.
2
Shift by one complete repeating block.
$100 x = 18.181818 \dots$
Multiplication by 100 aligns the recurring tails.
3
Subtract the equations.
$\begin{aligned} 100 x - x & = 18.181818 \dots - 0.181818 \dots \\ 99 x & = 18 \end{aligned}$
The identical infinite tails cancel exactly.
4
Solve and simplify.
$x = \frac{18}{99} = \frac{2}{11}$
This exact ratio of integers proves the decimal is rational.

Answer

Rational: $2 / 11$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Which is irrational: $\sqrt{81}$ , $0.125$ , $π$ or $- 4 / 7$ ?

Show answer & reasoning

$π$ .

Only $π$ is irrational. The other choices are $\sqrt{81} = 9 = \frac{9}{1}$ , $0.125 = \frac{1}{8}$ and $- \frac{4}{7}$ , all exact integer fractions.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Classify $- 6$ and support your answer with a fraction.
2 marks
Classify $0.375$ and write it as a simplified fraction.
2 marks
Classify $\sqrt{121}$ and explain.
2 marks
Classify $\sqrt{13}$ and explain.
2 marks
Write $0. \bar{27}$ as a simplified fraction.
3 marks
A student says $3.142$ is irrational because it is close to $π$ . Explain the error.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Classify $- 6$ and support your answer with a fraction.
2 marks
$- 6 = - \frac{6}{1}$ .
Rational.
Classify $0.375$ and write it as a simplified fraction.
2 marks
$0.375 = \frac{375}{1000} = \frac{3}{8}$ .
Rational; $\frac{3}{8}$ .
Classify $\sqrt{121}$ and explain.
2 marks
$121 = 11^{2}$ .
Rational; $11$ .
Classify $\sqrt{13}$ and explain.
2 marks
$13$ is a positive integer that is not a perfect square.
Irrational.
Write $0. \bar{27}$ as a simplified fraction.
3 marks
$100 x - x = 27$ , so $x = \frac{27}{99} = \frac{3}{11}$ .
$\frac{3}{11}$ .
A student says $3.142$ is irrational because it is close to $π$ . Explain the error.
3 marks
The decimal ends, so $3.142 = \frac{3142}{1000} = \frac{1571}{500}$ .
An approximation to an irrational value need not be irrational.
$3.142 = \frac{1571}{500}$ is rational.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 real-number classification and applications; root bounds and sensible digital approximation. Full surd manipulation is developed in Year 10.

Official curriculum source ↗

In this lesson

Understand the idea.

What makes a number rational?

Square roots and pi

See the method in action.

A root sign does not decide the answer

Simplify the perfect-square root.

Check whether 50 is a perfect square.

Classify the second root.

An exact recurring decimal

Name the recurring decimal.

Shift by one complete repeating block.

Subtract the equations.

Solve and simplify.

Try it yourself.

Check your understanding

Keep practising